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Approximations of the population Fisher information matrix- differences and consequences Joakim Nyberg, Sebastian Ueckert, Andrew C. Hooker. Background. At PODE 2009 all Population Optimal Design (OD) Software should evaluate the same simple Warfarin problem…

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Approximations of the population Fisher information matrix- differences and consequencesJoakim Nyberg, Sebastian Ueckert, Andrew C. Hooker

background
Background
  • At PODE 2009 all Population Optimal Design (OD)
  • Software should evaluate the same simple Warfarin problem…
  • 1-compartment model, 1st order absorption, oral dose 70 mg
  • Proportional error model (σ2=0.01)
  • 32 subjects with 8 measurements at

0.5, 1, 2, 6 ,24, 36, 72,120 hours (evaluation)

fisher information matrix fim
Fisher Information Matrix (FIM)

FIM can be calculated in different ways:

Assuming var(y) w.r.t. the fixed effects=0

Assuming var(y) w.r.t. the fixed effects≠0

A* is somewhat modified/updated if full is used, i.e.

Different between

full and reduced

fisher information matrix fim4
Fisher Information Matrix (FIM)

The FIM

  • If we have correlation between fixed effects
  • and random effects in like the FULL is the “theoretically
  • correct” method.
  • If not, the Reduced is “correct theoretically”

but this is seldom the case in Pharmacometrics

results from last pode 2009
Results from last PODE 2009

The “truth”

*

*

* Retout, Mentré – Further developments of FIM in NLME-models…. J. BioPharm. Stat 2003

results from last pode 20097
Results from last PODE 2009

Possibly issues with the

Cramér-Rao inequality

results from last pode 2009 summary
Results from last PODE 2009 summary
  • Software gave similar results with similar approximations
  • Reduced superior to Full in terms of predicting the “truth”
  • Even less predictive performance with higher order

FOCE-based FIM.

possible reasons initial ideas
Possible reasons – Initial ideas
  • The derivation of Full or Red is wrong
      • Derive FIM with simulations, i.e. integrate over observed FIM
  • FO-approximation too poor

- FOCE is obviously not enough, try high order approximations

  • Asymptotic behavior (FIM-1≤COV)

- Increase data set x 2 => SE should decrease by 2(1/2)

  • Numerical instability in Full but not in Red FIM

- Using automatic differentiation (AD) to avoid step length issues

  • Estimation software is not true ML-estimator,
  • i.e. efficiency of estimator not accurate
  • - NM hard to know how the parameter search is performed
  • but Monolix well documented
investigations reducing the complexity
Investigations – Reducing the complexity
  • ln-transform model to have additive res-error

(avoiding interaction terms)

  • Check that the problem holds for prop IIV structure

(FO approximation => proportional IIV = exp IIV)

  • Fix all parameters except fixed effect Ka
results reducing the complexity
Results – Reducing the complexity

ln model, add error, exp IIV = prop IIV

  • Issues still remaining => work with simplified model

* 100 000 bootstrap samples

** Retout, Mentré – Further developments of FIM in NLME-models…. J. BioPharm. Stat 2003

results full vs reduced
Results – Full vs Reduced
  • Asymptotic behavior (FIM-1≤COV)
      • Increase data set x 2 => SE should decrease by 2(1/2)
  • Numerical instability in Full but not in Red FIM

- Using automatic differentiation (AD) to avoid step length issues

None of this affected the results (2 down 3 to go)

next things to try
Next things to try...
  • The derivation of Full or Red is wrong
      • Derive FIM with simulations, i.e. integrate over observed FIM
  • FO-approximation too poor

- FOCE is obviously not enough, try high order approximations

SO

results high order approximations simulation based derivations

Similar

SO – Closer to truth

Simulation based FIM FOCE from NONMEM solve the problem!

Results – High order approximations & simulation based derivations

* 100 000 bootstrap samples

results full vs red
Results Full vs Red
  • The derivation of Full or Red is wrong
      • Integration FIM ≈ Analytic FIM => Not the answer
  • FO-approximation too poor
      • SO shrinks the differences but still to poor of an approx.
      • FOCE is worse but NONMEM integrated FOCE FIM is good?!
        • Possibly issues with the FOCE method?
foce fim differences improvements
FOCE FIM – Differences & Improvements
  • NONMEM FOCE assumes linearization around the mode
  • of the distribution => correlation between the individual
  • parameters and the population parameters.
  • Analytic FIMFOCE * does not assume this
      • To calculate individual mode data is needed
  • Update Analytic FIMFOCE to include the correlation:
  • Calculate Expected Empirical Bayes Estimates (EEBE)

EEBE are not data dependent

  • Whenever PopED differentiates pop parameters;

differentiate EEBE as well

* Retout, Mentré – Further developments of FIM in NLME-models…. J. BioPharm. Stat 2003

results updated fim foce
Results – Updated FIMFOCE
  • The new FOCE method solves the problem!
the answer
The answer
  • The Full FIM does not always work

with the FO-approximation

does full red affect the optimal design
Does Full/Red affect the optimal design?

Full

Reduced

6

2

2

2

1

1

1

1

3 support points

5 support points

does linearization method affect the optimal design
Does linearization method affect the optimal design?

Surface of |FIM| for SOCE-MC

SOCE>MC

SOCE=MC

SOCE<MC

Optimal MC

Optimal SOCE

Example from PAGE 2009 Nyberg et al

always use reduce fim instead
Always use reduce FIM instead?

Full |FIM|

Reduced |FIM|

Similar results with the transit compartment model

Results from Nyberg et al PODE 2008

surface of rse full fim only different from reduced in some regions
Surface of RSE(%) – Full FIM only different from reduced in some regions

Full Ka RSE(%)

Reduced Ka RSE(%)

S3-S7: (7.79h)

S3-S7: (7.79h)

S8: (120h)

S8: (120h)

Full ≈ Red

conclusions
Conclusions
  • The FO-approximation is not always enough for Full FIM

Possibly also too poor approx. for Reduced

  • Reduced FIM collapses occasionally
  • High order approximations stabilize differences
  • Different approximations give different optimal designs,

e.g. different sampling times and different number of

support points

suggestions
Suggestions
  • If runtime allows – Use high order approximations

FOCE, SO, SOCE, MC etc.

  • If Red is stable – Use reduced to optimize but evaluate with both
  • If Red is unstable – Optimize with Full but evaluate with Red

Beware: - No “golden” solution is presented

- The Cramer-Rao inequality does not hold comparing

different methods when optimizing / estimating

- To get “correct” SE from the FIM either sim/est needs

to be performed or high order FIMs need to be evaluated

thank you
Thank you

I would like to acknowledge Sergei Leonov for our interesting emails

discussing these issues.